Finding real roots of a Polynomial Equation without graphs.

Question

I am interested in finding the number of real roots of the polynomial equation

$$ x^9 + \frac{9}{8}x^6 + \frac{27}{64}x^3 - x + \frac{219}{512} = 0. $$

I know that graphing it would tell me how many real roots it has: the graph cuts the x-axis three times. But the coefficients are telling me some factorization is possible. I tried to write it like

$$ \left(x^3 + \frac{3}{8}\right)^3 = x - \frac{3}{8}, $$

but what next? Or is a graphical solution is the only possibility?

Answer 1

We have

$$ \begin{align} &x^9 + \frac{9}{8}x^6 + \frac{27}{64}x^3 - x + \frac{219}{512} \\ &\qquad = \frac{1}{512} (2 x-1) \left(4 x^2+2 x-3\right) \left(64 x^6+64 x^4+48 x^3+64 x^2+24 x+73\right). \end{align} $$

Since

$$ \begin{align} &64 x^6+64 x^4+48 x^3+64 x^2+24 x+73 \\ &\qquad \geq 64 x^6+48 x^3+24 x+73 \\ &\qquad > 28 x^6+48 x^3+24 x+44 \\ &\qquad = 4 (x+1)^2 \left(7 x^4-14 x^3+21 x^2-16 x+11\right) \\ &\qquad \geq 4 (x+1)^2 \left(7 x^4-14 x^3+15 x^2-16 x+8\right) \\ &\qquad = 4 (x+1)^2 (x-1)^2 \left(7 x^2+8\right) \\ &\qquad \geq 0, \end{align} $$

the only real roots are $x=1/2$ and

$$ x = \frac{-1 \pm \sqrt{13}}{4}. $$

Answer 2

The next level beyond Descartes' rule of signs is Sturm's theorem. Using it, you can find the number of real roots, or the number of roots in any interval, for any polynomial with real coefficients.

You start by finding the canonical Sturm sequence of the polynomial. In this case the Sturm sequence (normalized so the leading coefficients are $\pm 1$) is $$\pmatrix{{x}^{9}+{ {9\,{x}^{6}}/{8}}+{ {27\,{x}^{3}}/{64}}-x+{ { 219}/{512}} \cr{x}^{8}+3\,{x}^{5}/4+{ {9\,{x}^{2}}/{64}}-1/9\cr -{x}^{6}- 3\,{x}^{3}/4+{ {64\,x}/{27}}-{ {73}/{64}}\cr -x^3 + { {27\,{x}^{2} }/{64}}+{ {3}/{64}}\cr x^2-{ {1069962688\,x}/{175592043}}+{ {19822249}/{6503409}}\cr x -{ {44951085}{81718064}}\cr1 }$$
If $\sigma(a)$ and $\sigma(b)$ are the number of sign changes in the sequence at $a$ and at $b$ (where $a < b$, and $a$ and $b$ are not multiple roots of the polynomial), then the number of distinct roots in the interval $(a,b]$ is $\sigma(a) - \sigma(b)$.

For the total number of real roots, you just have to look at the leading terms of each polynomial in the sequence. Thus when $b \to +\infty$, the pattern of signs in the Sturm sequence is $+,+,-,-,+,+,+$: $\sigma(b) = 2$. When $a \to -\infty$, the pattern of signs is $-,+,-,+,+,-,+$: $\sigma(a)= 5$. $\sigma(a) - \sigma(b) = 3$, so there are three real roots.